38 
DR. R. S. BALL’S RESEARCHES IN THE DYNAMICS OE A RIGID 
*15 
7» 
ytfi— z A, 
z l a 1 x x y x , #ift y\U l 
* 2 , 
ft, 
7 2 , 
y iY-i ^ 2 ft, 
z 2 a 2 — x 2 y 2 , # 2 /3 2 — y 2 u 2 
*3, 
ft» 
735 
y$7z ^ft. 
Z 3 a 3 #3735 #sft ^3^3 
*4, 
ft, 
745 
y*y*-z&, 
Z\Cl 4 #4745 #ift ^4*4 
*55 
ft> 
755 
y*v 5 — z 5 ft, 
^5*5 #5755 #5^5 ^5*5 
*6, 
ft. 
765 
y&7& — z 6@6> 
Z 6 K 6 — #6765 #6ft y& a 6 
If « 15 j3 I5 y x be considered variable, all the other quantities remaining constant, we 
have the following theorem due to Mobius : — 
All the lines which can be drawn through a given point in involution with five given 
lines lie in a plane. 
This is in reality only a particular case of the following theorem, which appears from 
equating the general expression for the sexiant to zero : — 
All the screws of given pitch which can be drawn through a point so as to be coor- 
dinate with five given screws lie in a plane. 
A single screw X must be capable of being found which is reciprocal to all the six 
screws P, Q, E, S, T, U. Suppose the rigid body were only free to twist about X, then 
the six forces would not only collectively be in equilibrium, but severally would be 
unable to stir the body only free to twist about X. 
In general a body which was able to twist about six screws (of any pitch) would have 
perfect freedom ; but the body capable of rotating about each of the six lines P, Q, E, 
S, T, U, which are in involution, is not perfectly free, since practically we have only five 
disposable coordinates. 
If a rigid body were perfectly free, then a wrench about any screw could move the 
body ; if the body be only free to rotate about the six lines in involution, then a wrench 
about every screw (except X) can move it. 
The existence of the single screw X is the characteristic feature of six lines in invo- 
lution which the theory of screws makes known to us. 
The conjugate axes of Professor Sylvester (p. 743) are presented in the present system 
as follows : — Draw any cylindroid which contains the reciprocal screw X, then the two 
screws of zero pitch on this cylindroid are a pair of conjugate axes. For a force on 
any transversal intersecting this pair of screws is reciprocal to the cylindroid, and is 
therefore in involution with the original system. 
Draw any two cylindroids, each containing the reciprocal screw, then all the screws of 
the cylindroids form a coordinate system with three degrees of freedom [art. 84], 
Therefore the two pairs of conjugate axes, being four screws of zero pitch, must lie upon 
the same hyperboloid. This theorem is also due to Professor Sylvester. 
The cylindroid also presents in a very clear manner the solution of the problem of 
finding two rotations which shall bring a body from one position to any other given 
position. Find the twist which would effect the desired change. Draw any cylindroid 
